0 #

"

” ‘

are called row -permutation and column -permutation of , respectively.

d nd¡ £ ¨¡ d¥ $ ¥ £ ”¤§¤

u © n ¥ u§¥ 6¡

¤ §¥ ¡ ©

¨

¤©

¡ ¥ ©

˜ 51)1u`‘

˜

!

It is well known that any permutation of the set results from at most inter-

changes, i.e., elementary permutations in which only two entries have been interchanged.

An interchange matrix is the identity matrix with two of its rows interchanged. Denote by

H…

such matrices, with and being the numbers of the interchanged rows. Note that

0

P)11) 6 P )

in order to interchange rows and of a matrix , we only need to premultiply it by the

H… !

‘ ˜0

matrix . Let be an arbitrary permutation. This permutation is the

0

˜6r6 i1)1Y 0 C r' 0a ' wB

product of a sequence of consecutive interchanges . Then the X

rows of a matrix can be permuted by interchanging rows , then rows of the

a‘ ‘

resulting matrix, etc., and ¬nally by interchanging of the resulting matrix. Each of

0 £ "… £

these operations can be achieved by a premultiplication by . The same observation

can be made regarding the columns of a matrix: In order to interchange columns and of a

…p

matrix, postmultiply it by . The following proposition follows from these observations.

0

V¤ G S£

£¦ „

SRI FcP` `

Q

bQ

H QF T©

Let be a permutation resulting from the product of the inter-

' a ' rB ˜ 111)‘

changes , . Then,

X

C

¡ 0 ¡2 ¡ 2 ¡ " "

where

0 0 ¡

0 )1 ‘t

©¦ ¢ ¥

¨

"… ¤ "… "…

¤

0 )1 40 0 ¡ 0 ¨ ‘¢ ¥

t

"… "… "… &

3 3

Products of interchange matrices are called permutation matrices. Clearly, a permutation

matrix is nothing but the identity matrix with its rows (or columns) permuted.

… 6

Observe that , i.e., the square of an interchange matrix is the identity, or

#

"

0

equivalently, the inverse of an interchange matrix is equal to itself, a property which is

intuitively clear. It is easy to see that the matrices (3.1) and (3.2) satisfy

0 ¡ !¡ 0 1) 40 0 x 0 )1

0

"… ¤ "… "… "… "… "…

0 ¤

3 3

0

which shows that the two matrices and are nonsingular and that they are the inverse ¡ ¡

of one another. In other words, permuting the rows and the columns of a matrix, using

the same permutation, actually performs a similarity transformation. Another important

consequence arises because the products involved in the de¬nitions (3.1) and (3.2) of ¡

0 £ "… £

and occur in reverse order. Since each of the elementary matrices is symmetric, 0

¡

¡0

the matrix is the transpose of . Therefore, ¡

¡0 0 ¡

G¡ &

Since the inverse of the matrix is its own transpose, permutation matrices are unitary. ¡

Another way of deriving the above relationships is to express the permutation matrices

and in terms of the identity matrix, whose columns or rows are permuted. It can

¡ G¡

easily be seen (See Exercise 3) that

32 ¡ ¡2

" "

¡ G¡

It is then possible to verify directly that

¡ 2 ¡ 2 ¡ 2 ¡ G ¡

"

¡" 2

" "

¡ ud

¤§ ¥ §¢

¡

¡© © ¢ ¡ ¥ ¥ © ¡ ¥

§

It is important to interpret permutation operations for the linear systems to be solved.

When the rows of a matrix are permuted, the order in which the equations are written is

changed. On the other hand, when the columns are permuted, the unknowns are in effect

relabeled, or reordered.

¥ „ h

T © © §¦

¥ ¦

Consider, for example, the linear system where

‚‚ E667‚ 0 0 ‚ 34 @8 7

9

7 0

¡

‚ 7‚ 6 ‚ ‚ 5